Biomedical Engineering Reference
In-Depth Information
From Digital Filter to Transfer Function
The transfer function for the digital system,
H
(
z
), can be obtained by rearranging the dif-
ference equation (Eq.
(11.23)) and applying Eq.
(11.21).
(
)
is the quotient of
the
H
z
transform of the output,
(
), divided by the
transform of the input,
(
).
z-
Y
z
z-
X
z
y ð k Þþ a 1 y ð k
1
Þþ a 2 y ð k
2
Þ ...þ a N y ð k N Þ¼ b 0 x ð k Þþ b 1 x ð k
1
Þþ...þ b M x ð k M Þ
z 1 Y
z 2
Y ð z Þ ...þ a N z N Y ð z Þ¼ b
z 1
z 2
X ð z Þ ...: þ b M z M X ð z Þ
Y ð z Þþ a
ð
z
Þþ a
X ð z Þþ b
X ð z Þþ b
1
2
0
1
2
z 1
z 2
...þ a N z N Þ¼ X ð z Þð b
z 1
z 2
...: þ b M z M Þ
Y ð z Þð
1
þ a
þ a
þ b
þ b
1
2
0
1
2
X ð z Þ ¼ b 0 þ b 1 z 1
þ b 2 z 1
...þ b M z M
H ð z Þ¼ Y ð z Þ
þ a 1 z 1
þ a 2 z 1
...þ a N z N
1
ð
11
:
45
Þ
From Transfer Function to Frequency Response
The frequency response (
H 0 (
O
)) of a digital system can be calculated directly from
H
(
z
),
where
is in radians. If the data are samples of an analog signal as previously described,
the relationship between
O
o
and
O
is
O ¼ o
T:
H 0 ðOÞ¼ H ð z Þj z ¼ e j O
ð
11
:
46
Þ
For a linear system, an input sequence of the form
x
ð
k
Þ¼
A sin
ðO 0 k
þ FÞ
will generate an output whose steady-state sequence will fit into the following form:
y ð k Þ¼
B sin
ðO 0 k þ Þ
Values for B and
can be calculated directly:
¼ AH 0 ðO
B
j
Þ
j
0
ð H 0 ðO
¼ F þ
angle
ÞÞ
0
EXAMPLE PROBLEM 11.24
The input sequence for the digital filter used in Example Problem 11.23 is
p
2 k
x ð k Þ¼
100 sin
What is the steady-state form of the output?
Solution
1
2 y ð k
1
2 x ð k Þ
y ð k Þ
1
Þ¼
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